Near-Linear Algorithms for Geometric Hitting Sets and Set Covers

Abstract

Given a finite range space $$\Sigma =(\mathsf {X},\mathcal {R})$$, with $$N= |\mathsf {X}| + |\mathcal {R}|$$, we present two simple algorithms, based on the multiplicative-weight method, for computing a small-size hitting set or set cover of $$\Sigma $$. The first algorithm is a simpler variant of the Bronnimann–Goodrich algorithm but more efficient to implement, and the second algorithm can be viewed as solving a two-player zero-sum game. These algorithms, in conjunction with some standard geometric data structures, lead to near-linear algorithms for computing a small-size hitting set or set cover for a number of geometric range spaces. For example, they lead to $$O(N\mathrm {polylog}(N))$$ expected-time randomized O(1)-approximation algorithms for both hitting set and set cover if $$\mathsf {X}$$ is a set of points and $$\mathcal {R}$$ a set of disks in $$\mathbb {R}^2$$.